The Gowers uniformity norms of a (bounded, measurable, compactly supported) function taking values on a locally compact abelian group , equipped with a Haar measure ;

The Gowers uniformity norms of a function on a discrete interval ; and

The Gowers-Host-Kra seminorms of a function on an ergodic measure-preserving system .

These norms have been discussed in depth in previous blog posts, so I will just quickly review the definition of the first norm here (the other two (semi)norms are defined similarly). The norm is defined recursively by setting

and

where . Equivalently, one has

Informally, the Gowers uniformity norm measures the extent to which (the phase of ) behaves like a polynomial of degree less than . Indeed, if and is compact with normalised Haar measure , it is not difficult to show that is at most , with equality if and only if takes the form almost everywhere, where is a polynomial of degree less than (which means that for all ).

Our first result is to show that this result is robust, uniformly over all choices of group :

Theorem 1 (-near extremisers) Let be a compact abelian group with normalised Haar measure , and let be such that and for some and . Then there exists a polynomial of degree at most such that , where is bounded by a quantity that goes to zero as for fixed .

The quantity can be described effectively (it is of polynomial size in ), but we did not seek to optimise it here. This result was already known in the case of vector spaces over a fixed finite field (where it is essentially equivalent to the assertion that the property of being a polynomial of degree at most is locally testable); the extension to general groups turns out to fairly routine. The basic idea is to use the recursive structure of the Gowers norms, which tells us in particular that if is close to one, then is close to one for most , which by induction implies that is close to for some polynomials of degree at most and for most . (Actually, it is not difficult to use cocycle equations such as (when ) to upgrade “for most ” to “for all “.) To finish the job, one would like to express the as derivatives of a polynomial of degree at most . This turns out to be equivalent to requiring that the obey the cocycle equation

where is the translate of by . (In the paper, the sign conventions are reversed, so that , in order to be compatible with ergodic theory notation, but this makes no substantial difference to the arguments or results.) However, one does not quite get this right away; instead, by using some separation properties of polynomials, one can show the weaker statement that

where the are small real constants. To eliminate these constants, one exploits the trivial cohomology of the real line. From (1) one soon concludes that the obey the -cocycle equation

and an averaging argument then shows that is a -coboundary in the sense that

for some small scalar depending on . Subtracting from then gives the claim.

Similar results and arguments also hold for the and norms, which we will not detail here.

Dimensional analysis reveals that the norm is not actually the most natural norm with which to compare the norms against. An application of Young’s convolution inequality in fact reveals that one has the inequality

where is the critical exponent , without any compactness or normalisation hypothesis on the group and the Haar measure . This allows us to extend the norm to all of . There is then a stronger inverse theorem available:

Theorem 2 (-near extremisers) Let be a locally compact abelian group, and let be such that and for some and . Then there exists a coset of a compact open subgroup of , and a polynomial of degree at most such that .

Conversely, it is not difficult to show that equality in (2) is attained when takes the form as above. The main idea of proof is to use an inverse theorem for Young’s inequality due to Fournier to reduce matters to the case that was already established. An analogous result is also obtained for the norm on an ergodic system; but for technical reasons, the methods do not seem to apply easily to the norm. (This norm is essentially equivalent to the norm up to constants, with comparable to , but when working with near-extremisers, norms that are only equivalent up to constants can have quite different near-extremal behaviour.)

In the case when is a Euclidean group , it is possible to use the sharp Young inequality of Beckner and of Brascamp-Lieb to improve (2) somewhat. For instance, when , one has

with equality attained if and only if is a gaussian modulated by a quadratic polynomial phase. This additional gain of allows one to pinpoint the threshold for the previous near-extremiser results in the case of norms. For instance, by using the Host-Kra machinery of characteristic factors for the norm, combined with an explicit and concrete analysis of the -step nilsystems generated by that machinery, we can show that

whenever is a totally ergodic system and is orthogonal to all linear and quadratic eigenfunctions (which would otherwise form immediate counterexamples to the above inequality), with the factor being best possible. We can also establish analogous results for the and norms (using the inverse theorem of Ben Green and myself, in place of the Host-Kra machinery), although it is not clear to us whether the threshold remains best possible in this case.

As usual, the connection is easiest to state in a finite field model such as . In this case, we have the following inverse sumset theorem of Ruzsa:

Theorem 1. If is such that , then A can be covered by a translate of a subspace of of cardinality at most .

The constant has been improved for large in a sequence of papers, from by Ruzsa, by Green-Ruzsa, by Sanders, by Green and myself, and finally by Konyagin (private communication) which is sharp except for the precise value of the O() implied constant (as can be seen by considering the example when A consists of about 2K independent elements). However, it is conjectured that the polynomial loss can be removed entirely if one modifies the conclusion slightly:

Conjecture 1. (Polynomial Freiman-Ruzsa conjecture for .) If is such that , then A can be covered by translates of subspaces of of cardinality at most |A|.

Theorem 2. Let be a function whose norm is at least 1/K. Then there exists a quadratic polynomial such that .

Note that the quadratic phases are the only functions taking values in [-1,1] whose norm attains its maximal value of 1.

It is conjectured that the exponentially weak correlation here can be strengthened to a polynomial one:

Conjecture 2. (Polynomial inverse conjecture for the norm). Let be a function whose norm is at least 1/K. Then there exists a quadratic polynomial such that .

The first main result of this paper is

Theorem 3. Conjecture 1 and Conjecture 2 are equivalent.

This result was also independently observed by Shachar Lovett (private communication). We also establish an analogous result for the cyclic group , in which the notion of polynomial is replaced by that of a subexponential , and in which the notion of a quadratic polynomial is replaced by a 2-step nilsequence; the precise statement is a bit technical and will not be given here. We also observe a partial partial analogue of the correpsondence between inverse sumset theorems and Gowers norms in the higher order case, in particular observing that inverse theorems imply a certain rigidity result for “Freiman-quadratic polynomials” (a quadratic version of Conjecture 3 below).

I’ve just uploaded to the arXiv my paper “An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation“. This paper is another technical component of my “heatwave project“, which aims to establish the global regularity conjecture for energy-critical wave maps into hyperbolic space. I have been in the process of writing the final paper of that project, in which I will show that the only way singularities can form is if a special type of solution, known as an “almost periodic blowup solution”, exists. However, I recently discovered that the existing function space estimates that I was relying on for the large energy perturbation theory were not quite adequate, and in particular I needed a certain “inverse theorem” for a standard bilinear estimate which was not quite in the literature. The purpose of this paper is to establish that inverse theorem, which may also have some application to other nonlinear wave equations.

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